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photon density of order 10 photons per pixel (nearer 100 per pixel in the numerical experimentsdescribed here) and hence is inherently noisy.One way to reduce noise sensitivity is to model the emission process as a random processand to use reconstruction algorithms based on the statistical model. In 1982, Shepp and Vardi[SHEP82] introduced a Poisson process model for emission tomography which seems to be anexcellent model. Maximum likelihood (ML) and maximum a posteriori (MAP) methods basedon this model give what we consider to be the best possible reconstruction from noisy sinogramdata.The problem with the ML and MAP techniques is that they are very expensive in comparisonwith Fourier based techniques. The DFR method requires O(N2log N) operations, and CBPrequires O(N3) operations, where N is the number of detectors. On the other hand, an iterativeML algorithm (called the EM algorithm) requires 0(N4) operations for each iteration and itusually takes 50 iterations to achieve sufficient precision, [VARD85]. Even with accelerationtechniques such as Lewitt and Muehllehner's relaxation method, [LEWI86], the EM algorithmstill remains computationally very expensive.Another way to reduce noise sensitivity is to filter the sinogram data. Several filtering tech-niques for emission tomography are described in the literature. Linear low pass filters are easyto implement in connection with Fourier based methods, and are the current method of choice.A special lowpass filter called the parabola filter was developed at Washington University forthe purpose of PET noise reduction. The filter is called a "parabola" filter since the windowfunction is a quadratic function in the frequency domain (the filter is nonetheless a linear filter).The window function is the product of a triangle functionand a ramp functionu(w) = Sp~)wwhere Q is the cutoff frequency. Yang, [YANG91], implemented a noise filtering scheme usingprojection onto convex sets. Kuan et al., [KUAN85], developed an adaptive filter for PETimage restoration. According to Yang's simulation results, the parabola filter is among themost effective for use in connection with the CBP method, from the standpoint of subjectiveevaluation of the resulting reconstructed image. The underlying assumption for using a lowpassfilter such as the parabola filter is that the energy of a typical image is primarily concentrated inits low-frequency components and that the energy of a random noise is more spread out over thewhole frequency domain. While this assumption seems to be a reasonable one even for the noisein emission tomography, it is not clear to us that this assumption alone captures the significantfeatures of the noise present in emission tomography applications. In these applications, thenoise is signal-dependent Poisson type noise. Furthermore, while the gross structure of an imageappears as low-frequency components, sharp edges give rise to high-frequency components, andsmall details also give rise to high-frequency components. Therefore, the linear lowpass filtering